1995 AHSME Problems/Problem 12
Problem
Let be a linear function with the properties that
and
. Which of the following is true?
Solution 1
A linear function has the property that either for all
, or for all
. Since
,
. Since
,
. And if
for
, then
is a constant function. Since
,
Solution 2
If is a linear function, the statement
states that the slope of the line
is nonnegative: it is either positive or zero.
Similarly, the statement states that the slope of the line
is nonpositive: it is either negative or zero.
Since the slope of a linear function can only have one value, it must be zero, and thus the function is a constant. The statement tells us that the value of the constant is
, and thus that
. This leads to
Solution 3 (Common Sense)
It should be very clear that and
is contradictory because of the fact that linear functions are monotonic. The only thing that makes sense is
, and
. This means that
has a slope of
. So
. So
. Select
.
~hastapasta
See also
1995 AHSME (Problems • Answer Key • Resources) | ||
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Followed by Problem 13 | |
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